comparing analytical and numerical approaches to meteoroid … · 2020. 8. 7. · meteor...

Comparing analytical and numerical approaches to meteoroid orbit determinationusing Hayabusa telemetry

Trent JANSEN-STURGEON *, Eleanor K. SANSOM, and Philip A. BLAND

Department of Earth and Planetary Sciences, Curtin University, Bentley, Western Australia 6102, Australia*Corresponding author. E-mail: [email protected]

(Received 12 July 2017; revision accepted 21 July 2019)

Abstract–Fireball networks establish the trajectories of meteoritic material passing throughEarth’s atmosphere, from which they can derive pre-entry orbits. Triangulated atmospherictrajectory data require different orbit determination methods to those applied toobservational data beyond the Earth’s sphere of influence, such as telescopic observations ofasteroids. Currently, the vast majority of fireball networks determine and publish orbitaldata using an analytical approach, with little flexibility to include orbital perturbations.Here, we present a novel numerical technique for determining meteoroid orbits from fireballnetwork data and compare it to previously established methods. The re-entry of theHayabusa spacecraft, with its known pre-Earth orbit, provides a unique opportunity toperform this comparison as it was observed by fireball network cameras. As initial sightingsof the Hayabusa spacecraft and capsule were made at different altitudes, we are able toquantify the atmosphere’s influence on the determined pre-Earth orbit. Considering thesetrajectories independently, we found the orbits determined by the novel numerical approachto align closer to JAXA’s telemetry in both cases. Using simulations, we determine theatmospheric perturbation to become significant at ~90 km—higher than the firstobservations of typical meteorite dropping events. Using further simulations, we find themost substantial differences between techniques to occur at both low entry velocities andMoon passing trajectories. These regions of comparative divergence demonstrate the needfor perturbation inclusion within the chosen orbit determination algorithm.

INTRODUCTION

Fireball networks track meteoritic material as ittransits our atmosphere. Triangulated observations offireballs provide precise trajectories for these objects. Bypropagating such trajectories back in time, we canacquire orbital data for meteoroids, be it of cometary orasteroidal origin. For objects

dynamical history of the solar system, requires bothaccuracy and precision in the meteoroid’s initial orbitdetermination techniques.

One such analytical technique is outlined in section11 of the work by Ceplecha (1987), hereafter referred toas “C-87.” It includes two corrections to the initialvelocity vector based on simplifying assumptions todetermine the meteoroid’s pre-Earth orbit. An alternativeapproach would be a numerical propagation method—anintegration-based approach that iteratively propagates ameteoroid’s initial state vector, through the mostsignificant perturbations, back in time until the Earth’sinfluence is considered negligible, at which point the pre-Earth orbit is produced.

Historically, C-87 has long been used as the methodof choice due to its computational ease and convenience.However, as computational power has increased, so hasthe viability of the numerical approach. There are at leastnine groups that publish orbital data from meteor andfireball observations, and C-87 is used by all but one ofthem (C-87: Spurn�y et al. 2007; Madiedo and Trigo-Rodr�ıguez 2008; Brown et al. 2010; Gural 2011; Cookeand Moser 2012; Rudawska and Jenniskens 2014; Colaset al. 2015; Wi�sniewski et al. 2017; Numerical: Dmitrievet al. 2015). The current numerical approach used byDmitriev et al. (2015) is available as part of the stand-alone Meteor Toolkit package (hereafter referred to as“MT-15”) and will be compared alongside the novelnumerical propagation method described in this work.This new numerical method will hereafter be referred toas “JS-19.”

Some studies have been established in the pastcomparing the analytical and numerical approaches toorbit determination (Clark and Wiegert 2011); however,these comparisons were conducted using publishedmeteor observations with no pre-Earth sightings. Tocompare the various orbit determination methods, areal-world example with well-recorded data both beforeand after it encounters Earth’s perturbing influence,namely the pre-Earth orbit and the triangulatedatmospheric trajectory, respectively, would beinvaluable.

In November 2005, Japan Aerospace ExplorationAgency’s (JAXA’s) Hayabusa mission successfullyretrieved samples from the near-Earth asteroid 25143Itokawa (Nakamura et al. 2011). On its scheduled returnto Earth, the Hayabusa spacecraft made severaltrajectory correction maneuvers, the last being about 3days before predicted re-entry over the WoomeraProhibited Area (WPA), South Australia. Following thislast correction burn, the orbit was calculated using precisepositional telemetry by the Deep Space Network team atNASA’s Jet Propulsion Laboratory (Cassell et al. 2011).On June 13, 2010, 13:52 UT, the Hayabusa spacecraft

and its return capsule made a coordinated ballistic re-entry over WPA. This re-entry was recorded by twotemporary stations set up by JAXA’s ground observationteam (Fujita et al. 2011), four autonomous observatoriesof Australia’s Desert Fireball Network (Borovicka et al.2011), and one optical imaging station within NASA’sDC-8 airborne laboratory (Cassell et al. 2011). Althoughit is not strictly a meteoroid, the Hayabusa mission is afitting candidate for orbit determination analysis. Its re-entry mimicked real meteoroid entry phenomena in itsballistic nature and was observed in a similar fashion tofireballs, while also possessing a “ground truth” orbitfrom DSN telemetry.

METHODS

All orbit determination methods studied in thispaper utilize the same triangulated observation data andall return identical outputs, providing an excellentsetting for comparison and analysis.

The inputs are simply the meteoroid’s initial “state”taken at the highest reliable altitude that was observed.This initial state includes the absolute UTC time ofobservation (epoch time), and the triangulated positionand velocity vectors at this time, expressed in Earth-centered inertial (ECI) coordinates.

The outputs are the six classical orbital elements(COEs) that describe the original orbit of the meteoroidbefore the gravitational influence of the Earth/Moonsystem at the initial observed time, or epoch time. Theseorbital elements are the semimajor axis (a), eccentricity(e), inclination (i), argument of periapsis (x), longitudeof ascending node (Ω), and the true anomaly (h).However, the true anomaly is generally not quoted forentry orbits if the epoch time is provided.

In this method section, we will first review themethod C-87 outlined in Ceplecha (1987) by presentingthe approach in a more conceptual and modern setting,before going on to describe our new numerical method(JS-15). A detailed description of the Meteor Toolkit(MT-15) approach is given by Dmitriev et al. (2015).

Analytical Method of Ceplecha (C-87)

As first outlined in Ceplecha (1987), C-87 is based onthe assumption of an initial hyperbolic collision orbitwith Earth. Using the mathematical theory of conics, thehyperbolic entry orbit’s asymptote can be determined,which is taken to be the local path of the meteoroidaround the Sun before Earth’s gravitational influence, asshown in Fig. 1. There are two adjustments made to theinitial velocity vector that best estimate this local pathrelative to Earth. These adjustments are made to themagnitude and zenith angle of the initial velocity vector.

2150 T. Jansen-Sturgeon et al.

The magnitude adjustment to the initial velocityvector is twofold; first to account for the atmosphericinfluence, and second to account for Earth’s gravitationalattraction component. The pre-atmospheric velocity, v∞,can be determined using methods described in theappendix of Pecina and Ceplecha (1983, 1984). Using thisinertial pre-atmospheric velocity, v∞, and the escapevelocity at that particular height, vesc, the magnitude ofthe resulting geocentric velocity vector, vg, can bedetermined as follows:

vg ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikv1k2 � v2esc

qwhere vesc ¼

ffiffiffiffiffiffiffiffiffiffiffi2 � lekx0k

s(1)

where le = G 9 me = 3.986005 9 1014 m3 s�2 (Moritz

2000) is Earth’s standard gravitational parameter, andx0 is the inertial position corresponding to the highesttriangulated point.

The direction of the geocentric velocity vector issimply the direction of v∞ with an adjustment to itszenith angle, zc, as follows:

ag ¼ ac; zg ¼ zc þ dzcwhere dzc ¼ 2 � tan�1 kv1k � vgkv1k þ vg tan

zc2

� �� (2)

where ac and zc are the local azimuth and zenith anglesof the observed radiant, corrected for Earth’s rotation,and ag and zg are the azimuth and zenith angles of thegeocentric radiant.

The geocentric velocity vector can now bedetermined from the velocity’s magnitude, azimuth, and

zenith angles formulated above. The resulting orbit isthen calculated by transforming the geocentric positionand velocity vectors, x0 and vg, into heliocentric inertialcoordinates (J2000) followed by heliocentric COEs.Notice there are no modifications to the position of themeteoroid due to Earth’s influence, or any otherperturbing body, as it is assumed that any adjustmentwould make near negligible difference to the resultingorbital elements.

We must note that C-87 cannot determine the orbitof an entry object that had been gravitationally boundto Earth due to its primary assumption of an initialhyperbolic collision orbit with Earth. We must also notethat the determination of the pre-atmospheric velocity,v∞, as outlined in the appendix of Pecina and Ceplecha(1983), is not as well known among the meteormodeling community, and is frequently mistaken as thevelocity at the first triangulated point, v0 (Vida et al.2018). This will lead to a misuse of Equations 1 and 2in calculating vg as the Earth’s atmospheric influencewill not be accounted for. In order to objectively assessthe effect of omitting the pre-atmospheric velocitydetermination on orbital calculations, we willadditionally analyze C-87 setting v∞ to v0.

Novel Numerical Method (JS-19)

Unlike C-87, JS-19 makes no assumptions about theorigin of the meteoroid and can accommodateperturbations with ease. This method effectively rewindsthe clock by propagating the meteoroid’s state back intime to a point well outside the Earth’s sphere ofinfluence (SOI).

Fig. 1. Ceplecha’s orbital assumption in the local region of Earth, where vg is the uninfluenced pre-Earth velocity vector and v∞ isthe (Earth affected) velocity vector determined outside the atmosphere using the method described in the appendix of Pecina andCeplecha (1983, 1984). [Globe image credit: earthobservatory.nasa.gov] (Color figure can be viewed at wileyonlinelibrary.com.)

Comparing meteoroid orbit determination methods 2151

www.wileyonlinelibrary.com

Modified equinoctial orbital elements (EOEs) areused to describe the meteoroid’s state as these elementsavoid the singularities inherent in the COEparameterization at zero- and ninety-degree inclinationsand zero eccentricity (Cefola 1972; Betts 2000). Theinitial conditions, namely the highest reliable inertialposition, x∞, and velocity, v∞, are converted frominertial vector coordinates into COEs and then fromCOEs into EOEs, as outlined in the Determining theorbital elements and Modified Equinoctial OrbitalElements, sections, respectively, of Colasurdo (2006).These EOEs are vectorized following the EuropeanSpace Agency’s notation (Walker et al. 1985) as:

y ¼ ½p; f; g; h; k;L�T (3)

In order to propagate the meteoroids state elementsback to its originating orbit, a dynamic model (or a setof ordinary differential equations) is needed, namely thevariation of parameters on the equinoctial elementmodel (Betts 2000):

_y ¼ A � utot þ b (4)

b ¼ 0 0 0 0 0 ffiffiffiffiffiffiffiffiffiffiffile � pp w=pð Þ2h iT (6)where A is the state rate matrix, b the state rateconstant, and utot is the total perturbing acceleration inthe body frame [radial, tangential, normal]. Also w, s, r,and b are some shorthand notations of commonexpressions:

w ¼ 1þ f � cosðLÞ þ g � sinðLÞ; s2 ¼ 1þ h2 þ k2r ¼ p=w; b ¼ h � sinðLÞ � k � cosðLÞ: (7)

For accurately determining the original orbit ofincoming meteoroids, perturbations need to be added tothis dynamic model. However, as there will be relatively

minimal net movement of the meteoroid through time,the only perturbations that would non-negligibly affectthe resulting orbit are those produced by the Earth/Moon system. These include the atmospheric drag, andthird body gravitational and zonal harmonicperturbations.

The first zonal harmonic (J2) perturbation is dueto the Earth’s oblate shape, and is about three timesthe magnitude of the next zonal harmonic (Moritz2000). Therefore, the Earth’s J2 zonal harmonicperturbation is the only one considered, and iscalculated in the body frame as follows (Kechichian1997):

uJ2 ¼�3 �le � J2 �R2e

r4 � s4s4� 12 �b2� �=2

4 �b � h � cos Lð Þþk � sin Lð Þð Þ2 �b � 2� s2� �

24

35 (8)

where J2 = 1.08263 9 10�3 (Moritz 2000) is the

dynamical form factor of the Earth, and Re = 6371.0 km(Moritz 2000) is the Earth’s mean radius.

The Newtonian third body perturbationequation has been shown to often promote substantialnumerical errors due to the significantly different

magnitude of the terms involved (Battin 1999). Toavoid this numerical inaccuracy, the followingequation (Betts and Erb 2003) is used to model thirdbody perturbations in the inertial (J2000) frame:

utb ¼�ltbxm þ f � qtbkxm � qtbk

where f¼ 3 � qþ 3 � q2 þ q3

1þ 1þ qð Þ3=2and q¼ xm � xm � 2 � qtbkqtbk2

(9)

where xm is the position of the meteoroid, qtb theposition of the third body, and ltb the standardgravitational parameter of the third body.

A ¼

_p_f

_g_h_k

_L

2666666664

3777777775¼ 1

w

ffiffiffiffiffip

le

r0 2 � p 0

w � sin Lð Þ wþ 1ð Þ � cos Lð Þ þ f �g � bw � cos Lð Þ wþ 1ð Þ � sin Lð Þ þ g f � b

0 0 s2 � cos Lð Þ=20 0 s2 � sin Lð Þ=20 0 b

2666666664

3777777775

(5)


Finally, while the atmospheric drag acceleration is afairly standard formula, the density of air in the upperatmosphere is not. The density in this region varies withnot only height, but latitude, longitude, time, and solaractivity. To incorporate all these subtle effects, weutilized the empirical NRLMSISE-00 atmosphericmodel (Picone et al. 2002) to calculate the atmosphericdensity (qair) within our drag equation:

udrag ¼ �qair � Cd � S � kvrelk � vrel2 �M (10)

where M is the mass of the meteoroid, Cd the dragcoefficient, S the meteoroid’s cross-sectional area, andvrel the meteoroid’s velocity vector relative to thesurrounding atmospheric air. Note that like the thirdbody perturbation, the atmospheric drag perturbationneeds a coordinate transformation into the body frameto be used in the dynamic model.

Now that the dynamic model is established(Equations 3–10), a numerical integrator is needed topropagate the meteoroid’s state variables through time.We have chosen a Runge–Kutta Dormand–Prince(RKDP) (Dormand and Prince 1980) method for theintegration due to its ability to constrain relative errorsby internally controlling step size—a new approach tonumerical fireball orbit modeling. Additionally, itsupports a good accuracy to computation ratio, namelyfifth-order accuracy for six function evaluations per step.

The RKDP method computes and compares afourth- and fifth-order Runge-Kutta solution in parallelto determine whether the current time step is sufficientlysmall. If the difference between the solutions exceeds theerror bounds, then the time step is decreased (by 1/10)and the RKDP is rerun on the current iteration step. Ifthis difference is much smaller than the error bounds,the current solution is taken and the time step isincreased (by 1/10) for the next RKDP iteration. Thecoefficients of the RKDP were chosen to minimize theerror of the fifth-order solution; therefore, it is thissolution that is used in the next step of the integrationprocedure.

Starting with an initial step size estimate of a tenth ofa second, we use the RKDP iterative integration processto propagate the meteoroid’s ECI EOE’s, yinitial, to theedge of the Earth’s SOI, where the coordinates areconverted into the Sun centered inertial frame (J2000).The integration process is then continued until themeteoroid has propagated to 10 SOI, upon which theEarth/Moon perturbations are removed from thedynamic model and the meteoroid is propagated back toepoch time. The resulting orbital elements, yfinal, reflectthe meteoroid’s original orbit around the Sun expressedin J2000 coordinates, and can be trivially converted toCOEs as described in section 3.4 of Colasurdo (2006).

Discontinuities can arise when switching betweengeocentric and heliocentric reference frames. To avoidsuch a discontinuity at the limit of Earth’s SOI, the Sunand Moon are considered perturbations when in thegeocentric frame, while the Earth and Moon areconsidered perturbations within the heliocentric frame.

The JS-19 method described above is similar to thatof MT-15 (Dmitriev et al. 2015), but differs in thechoice of state representation, integration method, anderror handling.

RESULTS AND DISCUSSION

To properly compare these methods for theiraccuracy, an example object with both a measured orbitand a measured bright flight entry would be invaluablefor analysis. The re-entry of the Hayabusa missionconstitutes an excellent calibration event in this regard,with a measured pre-Earth rendezvous orbit, asdetermined by the spacecraft’s navigational systems, andan observed re-entry trajectory, as published inBorovicka et al. (2011).

JAXA’s engineering team kindly provided theirorbital telemetry data for the Hayabusa mission (throughpersonal communication) at 2010-06-09T06:04:00.0UTC, just after its final correction maneuver (TCM-4), inthe form of a J2000 equatorial (Earth-Centered Inertial)state vector. This position and velocity state is easilyconverted into the following COEs:

Rx

Ry

Rz

Vx

Vy

Vz

2666666664

3777777775ECI

¼

�1:074047355� 106 km1:232756795� 106 km0:935509892� 106 km2:751442755 km s�1

�3:231296260 km s�1�12:442756954 km s�1

2666666664

3777777775)

a

e

i

x

X

h

2666666664

3777777775¼

1:32381AU

0:25732

1:68383�

147:47773�

82:46569�

27:71277�

2666666664

3777777775

(11)


Triangulated positions of the Hayabusa re-entryfrom ground-based observations are detailed inBorovicka et al. (2011). Two reduced trajectories given inthis work are for the observed re-entry of the spacecraftand for the capsule; these can be used as two separatecases for orbit determination method comparisons. Thetabulated triangulated positions and time in Borovickaet al. (2011) are used to determine the velocity, therebydefining the initial conditions of the luminous trajectories(see the Appendix). In both the spacecraft and capsulecases, the numerical propagation methods will integratethe corresponding object back to the time of telemetryreading for consistent orbital comparisons. Since C-87does not consider any perturbations, an epoch changewould simply require a two-body propagation, alteringonly the orbit’s anomaly (h). As this sixth element is notneeded for orbit comparison analysis, the epochrecalculation is not necessary.

In the two cases, we compare the orbit determinedusing the three different methods (C-87, MT-15, andJS-19). As the atmospheric correction to the initial velocityvector (Pecina and Ceplecha 1983, 1984) used in C-87 isnot always applied (Vida et al. 2018), we shall additionallyprovide the orbital results of C-87 by equating v∞ and v0(see the Methods section) to assess the effects.

Hayabusa’s Orbit Determined from the Spacecraft’sRe-entry

The initial position vector and corresponding initialtime of the spacecraft can be taken directly from table 2of Borovicka et al. (2011) at a height of 99.88 km.However, as there was no given radiant vectordescribing the spacecraft’s velocity, the initial velocityvector of the spacecraft was deduced using a straightline least squares approach on the first three1

triangulated positional data points with timing in table2 of Borovicka et al. (2011).

Additionally, the atmospheric perturbation modelrequires an estimated mass and cross-sectional area ofthe object to more accurately model the aerodynamics.While the mass and shape of the spacecraft arerelatively well documented to be 415 kg and1.5 m 9 1.5 m 9 1.05 m cube, respectively, theorientation of the spacecraft with respect to theatmosphere is more uncertain. This leaves us to assumethat the spacecraft’s cross-sectional area corresponds toits most aerodynamically stable orientation.

Using these initial conditions, the heliocentric orbitis calculated using all three methods and is compared tothe orbit derived from the spacecraft’s navigationsystem (Table 1; Fig. 2). The Southworth and Hawkins(1963) similarity criterion is included in Table 1 as aquantitative measure of the orbital difference betweenJAXA’s telemetric orbit and the orbit determined usingthe respective methods.

The perturbations included in JS-19 are those ofthe Earth, Moon, and Sun’s gravity; the Earth’s firstzonal harmonic (J2); and the atmospheric drag. Theseare the nonreversible, significant perturbing effects.Their respective strengths are calculated iterativelyduring the backward integration of JS-19 and arerepresented in Fig. 3. C-87 neglects the majority ofthese influences.

Hayabusa’s Orbit Determined from the Capsule’sRe-entry

The second interesting case is that of the Hayabusacapsule’s re-entry; it is only distinguished from the otherparts of ablating spacecraft much lower down in theatmosphere (~65 km altitude). Although the capsule hasalready decelerated heavily by this point, its mass andcross-sectional diameter are very well documented to be20 kg and 40 cm, respectively. This sets us up for anexcellent comparative study as to the effects of the

Table 1. The calculated heliocentric classical orbital elements for the Hayabusa satellite’s Earth rendezvous ascompared to the telemetric orbital data at T = 2010-06-09T06:04:00.0 UTC.

Heliocentric orbital elements

(ECLIPJ2000)

Telemetry

dataaC-87

(Ceplecha 1987)

C-87 (v∞ = v0)b

(Ceplecha 1987)

MT-15

(Dmitriev et al. 2015)

JS-19

(this work)

a (AU) 1.32381 1.30395 � 0.003 1.32000 � 0.003 1.32241 � 0.001 1.32265 � 0.003e 0.25732 0.24589 � 0.002 0.25472 � 0.002 0.25646 � 0.0007 0.25654 � 0.002i (°) 1.68383 1.64028 � 0.007 1.67009 � 0.007 1.68203 � 0.002 1.68367 � 0.007x (°) 147.47773 147.96599 � 0.2 147.67417 � 0.2 147.48000 � 0.07 147.52451 � 0.2Ω 82.46569 82.34476 � 0.001 82.34414 � 0.001 82.46687 � 0.0002 82.46664 � 0.002Similarity criterionc N/A 0.01178 0.00269 0.00087 0.00082

The errors are determined assuming 10 m s�1 error on the initial velocity magnitude, as discussed in the Precision section.aObtained through private communication with JAXA’s engineering team.bThe pre-encounter velocity, v∞, uses the velocity calculated at first observed point (v0). See the Methods section for details.cSouthworth and Hawkins (1963) similarity criterion as compared to the telemetry data.

1The orbits derived by fitting different numbers of initial data points

were analyzed and compared with similar results. For simplicity, only

one case is documented in this paper.


atmospheric perturbation on the resulting orbitalpredictions.

The initial inputs for this case originated from table3 of Borovicka et al. (2011). The initial positioncorresponds to the highest recorded sighting of thecapsule with timing, corresponding to 64.71 km. Theinitial velocity vector is deduced from the first two givenposition data points with timing. Note a straight line

least squares fit was not attempted here as the capsulewas already in a state of high deceleration. Also note thatthe telemetry provided for the capsule is approximated aswe have not accounted for the small delta-v used incapsule ejection 3 h prior to re-entry (Cassell et al. 2011).Again, two cases of C-87 are assessed using differentinitial velocity approaches. The comparison of orbitalresults is shown in Table 2 and Fig. 4.

Fig. 2. A comparison between Hayabusa’s heliocentric orbit (as determined from telemetry) and the spacecraft orbit calculated usingCeplecha’s analytical method (C-87), Dmitriev’s numerical method (MT-15), and the new numerical method outlined in this work (JS-19), as projected on the plane of the ecliptic. The comparison also features the inner terrestrial planets as references. Included is anenlarged view around the communal aphelion to emphasize the orbital discrepancies. Note the difference in CAM orbits usingdifferent pre-atmospheric velocity assumptions (see the Methods section). (Color figure can be viewed at wileyonlinelibrary.com.)

Fig. 3. Selected perturbations over the Hayabusa’s orbit from the final correction maneuver (TCM4) until Earth rendezvous.Note: the Earth’s J2 and atmospheric drag perturbations are considered negligible outside the Earth’s SOI at 924,000 km, andabove the exosphere at 10,000 km, respectively. (Color figure can be viewed at wileyonlinelibrary.com.)


www.wileyonlinelibrary.comwww.wileyonlinelibrary.com

The drastic difference between the predicted orbitsof the Hayabusa capsule is primarily due to how theatmosphere is considered by the various methods. Thisorbital discrepancy really highlights the importance of awell-modeled atmospheric perturbation influence in theorbit determination algorithm, especially for thoseobjects initially observed at lower altitudes, such assome meteorite dropping fireballs.

Atmospheric Influence

The significant difference between the determinedpre-Earth orbits of the Hayabusa capsule is due to thehandling of perturbations. The most dominantperturbation in this case is the atmosphere. To assess the

altitude at which the atmospheric influence on theorbit diminishes, we compared JS-19 (that accountsfor the atmosphere) to C-87 where v∞ = v0 (thatnegates the atmosphere). The initial conditions forthese comparisons were determined using JS-19; that is,the Hayabusa capsule was integrated back along its re-entry path to a specified altitude at which point C-87(v∞ = v0) was initiated alongside JS-19. The orbitaldifference between these two orbit determinationmethods from these initiation altitudes was thendetermined using the Southworth and Hawkins (1963)similarity criterion.

Figure 5 reveals a couple of interesting features aboutthe comparative nature of the two orbit determinationmethods. First, the similarity is shown to converge to a

Table 2. The calculated heliocentric classical orbital elements for the Hayabusa capsule’s Earth rendezvous ascompared to the telemetric orbital data at T = 2010-06-09T06:04:00.0 UTC.

Heliocentric orbitalelements (ECLIPJ2000)

Telemetrydataa

C-87(Ceplecha 1987)

C-87 (v∞ = v0)b

(Ceplecha 1987)MT-15(Dmitriev et al. 2015)

JS-19(this work)

a (AU) 1.32381 1.38633 � 0.003 1.17873 � 0.003 1.36699 � 0.001 1.31322 � 0.003e 0.25732 0.28928 � 0.002 0.16954 � 0.002 0.27995 � 0.0007 0.25160 � 0.002i (°) 1.68383 1.75327 � 0.007 1.32041 � 0.007 1.73243 � 0.002 1.64657 � 0.007x (°) 147.47773 150.05468 � 0.2 138.57245 � 0.2 149.13093 � 0.06 146.99422 � 0.2Ω (°) 82.46569 82.34249 � 0.001 82.35312 � 0.001 82.44881 � 0.0002 82.47087 � 0.002Similarity criterionc N/A 0.03413 0.09428 0.02394 0.00615

The errors are determined assuming a 10 m s�1 error on the initial velocity magnitude, as discussed in the Precision section.aObtained through private communication with JAXA’s engineering team.bThe pre-encounter velocity, v∞, uses the velocity calculated at first observed point (v0). See the Methods section for details.cSouthworth and Hawkins (1963) similarity criterion as compared to the telemetry data.

Fig. 4. A comparison between Hayabusa’s orbit (as determined from telemetry) and the capsule’s orbit calculated by Ceplecha’sanalytical method (C-87), Dmitriev’s numerical method (MT-15), and the new numerical method outlined in this work (JS-19),as projected on the plane of the ecliptic. Included is an enlarged view around the communal aphelion to emphasize the orbitaldiscrepancies. Note the difference in C-87 orbits using different pre-atmospheric velocity assumptions (see the Methods section).(Color figure can be viewed at wileyonlinelibrary.com.)



fixed value ~90 km altitude, indicating the atmosphericinfluence on the orbit diminishes at this point. Manymeteorite dropping events are not observed before thisaltitude, and are thus already experiencing significantatmospheric drag. However, the object’s physicalcharacteristics, such as mass, shape, and density, woulddirectly influence the magnitude of this atmosphericperturbation, and hence, this 90 km convergence altitude isspecific to the case of the Hayabusa capsule. Variation tothis altitude for other events requires further investigation.

Second, the apparent asymptote at high altitudes isnonzero. This is due to the continuing effects of thelarger scale perturbations acting on the object, namelythe Earth flattening and third body effects. While themagnitude of the Earth flattening perturbation dropsoff relatively quickly, the third body perturbationscontinue to influence the object’s orbit over theduration of the integration.

Error Analysis

For the orbital results to be validated and properlycompared, their errors must be identified andquantified. These errors originate from a variety ofsources, which can be factored into two groups—theobservational errors and the model errors.

The observational errors are simply theuncertainties associated with the epoch time, the initialtriangulated position vector, and the initial determinedvelocity vector before the orbital calculations begin.While the epoch time and positional errors are merelythe uncertainties in the measurement data, the velocityerrors are not so straightforward. The directional errorsof the velocity are calculated by considering thetriangulated positional radiant data as a whole,therefore minimizing the potential errors in the radiantentry angle. The errors in velocity magnitude aredetermined by referring to the velocity scatter at the

beginning of the object’s observable bright flight,before the atmosphere presents a significant resistiveinfluence.

The model errors are the uncertainties introducedwithin the orbit determination method itself, such as theimperfect nature of the state equations in representingmeteoroid flight (small perturbations missed, etc.),performing discrete time integration using the Dormand–Prince integrator (bounded at 1 mm per time-step), andthe use of coordinate transforms.2 Despite modeluncertainties being small with respect to observationalerrors, their inclusion must be considered for a robustanalysis. Combining all these uncertainties gives theoverall error, or precision, of the results.

PrecisionThe precision of the orbit determination methods is

primarily controlled by the error in the initial velocitymagnitude.3 The epoch time error, initial triangulatedposition error, and the model errors combined cause anorbital uncertainty three orders of magnitude smallerthan the initial velocity magnitude error alone. Theinitial velocity directional error is somewhat moreinfluential on the resulting orbital errors, but stillbetween one and two orders of magnitude smaller thanthe orbital uncertainty caused by the initial velocitymagnitude error.

No individual position errors were provided for thetriangulation results in the original paper (Borovickaet al. 2011). In order to perform a general error analysis,a velocity magnitude error of 10 m s�1 was assumed,with error results given alongside the correspondingorbital elements in Tables 1 and 2. Other velocity

Fig. 5. Orbital similarity between C-87 (v∞ = v0) and the new numerical approach at different initial altitude states, according tothe Southworth and Hawkins (1963) similarity criterion. (Color figure can be viewed at wileyonlinelibrary.com.)

2All coordinate transforms were performed using version 1.3 of

Python’s astropy module.3Using typical errors calculated by atmospheric trajectory modeling.



magnitude errors were considered and found to scaleroughly linearly to the resulting orbital errors; that is,multiplying the velocity magnitude error by two causesthe orbit uncertainty to double.

The errors on JS-19 are calculated using a Monte Carloapproach to handle the nonlinearity of the includedperturbations, where the error on the initial velocitymagnitude can be transformed into errors on the finalorbital elements. The reliability of these errors wasconfirmed through repeated Monte Carlo trials eachconsisting of 1000 particles. The error on the orbitdetermined by C-87 was also calculated using a MonteCarlo approach; however, the error determined by MT-15uses a series of covariance transforms throughout thealgorithm. This covariance approach linearizes the error ateach step, and therefore does not account for any significantnonlinear effects, such as a close encounter with the moon.

Tables 1 and 2 reveal that the orbital precisions ofC-87 and JS-19 only differ significantly in their longitudeof ascending node, Ω. This small discrepancy is due to C-87 assuming that the meteoroid’s original (pre-perturbed)Ω is simply the Earth’s heliocentric longitude at the timeof initial contact, which does not completely account forthe Earth’s gravitational influence on the meteoroid’strajectory. Clark and Wiegert (2011) suggest that “thevery tight uncertainties often reported for Ω are far tooaggressive, and should be minimally expanded toincorporate this discrepancy.” This is clearlydemonstrated by comparing the true Ω to the analyticallyand numerically determined Ω in Tables 1 and 2,highlighting the imprecise assumption that C-87 employs.

AccuracyWhile the precision describes the spread of orbital

results around the determined solution, the accuracy is ameasure of how close that solution comes to the trueorbit, or in our case, the orbit as determined using thespacecraft’s navigational systems. This error can bequantified by calculating the difference between the trueorbital elements and the determined orbital elements.However, a more robust and encompassing measure ofthe determination method’s accuracy is by employingthe similarity criterion (Southworth and Hawkins 1963).As shown in Tables 1 and 2, the new numericalapproach consistently produces more accurate orbitalresults. This comparison of accuracy has also beendemonstrated visually in Figs. 2 and 4.

Relative Similarity

A further assessment of similarity between C-87 andJS-19 can be made beyond the single observedHayabusa re-entry using a variety of simulated re-entrytrajectories. We can generate simulated trajectories using

the Earth fixed re-entry radiant unit vector of theHayabusa satellite as the trajectory backbone. This isthen varied by artificially altering the velocity magnitudeand the time of re-entry. By modifying the re-entry time,we are effectively adjusting the longitude of the re-entryin an inertial frame due to the Earth’s diurnal rotation.We vary the re-entry time through an entire day in20 min increments, given in UTC time. At each of thesediscrete time increments, the re-entry velocity magnitudeis also varied to cover all possible heliocentric orbitsconservatively, that is, from 10 up to 80 km s�1 in250 m s�1 increments; any resulting hyperbolic orbits aredismissed. On each of the 2088 simulated trajectorieswithin this data set, the orbit is computed once using C-87 and once using JS-19. The similarities of thedetermined heliocentric orbits are shown in Fig. 6.

The general shape of Fig. 6 is due to the Earth’svelocity around the Sun. At about 15:00 UTC on June13, 2010, the Earth’s velocity acts in the same direction asthe simulated Hayabusa re-entry, therefore reducing thevelocity needed to obtain a hyperbolic orbit relative toEarth. Conversely, around 03:00 UTC, the simulatedvelocity relative to Earth must be much higher to obtain ahyperbolic orbit as the Earth’s velocity opposes thesimulated Hayabusa re-entry velocity. Additionally, theminimum Earth centered velocity needed to obtain aheliocentric orbit is the Earth’s escape velocity, regardlessof the Earth’s orientation around the Sun.

Interestingly, certain regions of orbital dissimilaritycan be identified by excluding particular perturbationsfrom JS-19. For example, by removing the Moon’sgravitational perturbing influence, the orbit produced bythe numerical algorithm becomes more like the orbitproduced by C-87 in the area between 08:00 and 12:00UTC (Fig. 7).

Other regions of orbital dissimilarity in Figs. 6 and7 can also be identified. The darker region at lower re-entry velocities is due to the resulting orbit being closeto that of the Earth’s orbit, and therefore experiencing agreater time for the Earth/Moon perturbations toinfluence the orbit off its Keplerian path. Additionally,the roughly horizontal region at higher re-entryvelocities, around 05:00 UTC, corresponds to an area ofhigh orbital eccentricity (Fig. 8). As Jopek (1993)describes, the values of the similarity criterion(Southworth and Hawkins 1963) “strongly depend onthe orbit eccentricity when e > 0.9,” thereforeaccounting for this region of apparent dissimilarity.

Also note, the isolated dots in Figs. 6 and 7 are singleorbital cases where the inclination is so close to zero thatthe calculated longitude of ascending node, Ω, in oneorbital estimation is the longitude of descending node, ℧,in the other. This results in a misdiagnosis of orbitalsimilarity.


To give the reader an idea of these orbitaldifferences, if the velocity magnitude uncertainty was theonly acting source of orbit error, then 1, 10, 100 m s�1

uncertainties on initial velocity magnitude wouldcorrespond to orbital similarities of 0.0002, 0.002, and0.02, respectively. These correlations are specific to thegeometry of the Hayabusa trajectory and may vary for

different events, but serve well as a rough similarityconversion for Figs. 6 and 7. That said, the similarityrange in Figs. 6 and 7 are capped at 0.001 to highlightsubtleties; however, some orbit comparisons, especially atthe low velocity end, did show similarities on the orderof 0.02, or roughly 100 m s�1 variation in initial velocitymagnitude—a significant difference in orbital terms.

Fig. 7. Orbital similarity (Southworth and Hawkins 1963) between Ceplecha’s analytical method (C-87) and the new numericalmethod described in this paper (JS-19) having removed the Moon’s perturbation influence from the latter. Note the removal ofthe lunar effect between 08:00 and 12:00 UTC from Fig. 6. (Color figure can be viewed at wileyonlinelibrary.com.)

Fig. 6. Orbital similarity between Ceplecha’s analytical method (C-87) and the new numerical method described in this paper (JS-19) according to the Southworth and Hawkins (1963) similarity criterion. The darker the shade, the more difference there is betweenthe simulated orbits. Only the heliocentric orbits are shown; all hyperbolic and geocentric orbits are discarded. The sinusoidal-likeshape is due to the orbital velocity of the Earth around the Sun. The two distinctly darker areas at lower velocities represent strongperturbations that are not considered in the C-87 model. (Color figure can be viewed at wileyonlinelibrary.com.)



This analysis highlights differences in orbitdetermination methods due to re-entry timing andvelocity magnitude, and even further differences may becaused by variations in re-entry height, latitude, andazimuth and zenith angles.

So, without the ability to include perturbations, C-87cannot properly account for the complexities inherent inthe estimation of pre-Earth orbits. Any discrepancy fromthe meteoroid’s “true” orbit will be magnified when aprobabilistic method, such as Bottke et al. (2002), is usedto determine its orbital origins, therefore making itsignificantly harder to link meteoroids to their rightfulparent bodies or source regions.

CONCLUSIONS

Ceplecha’s analytical method of orbitdetermination (C-87; Ceplecha 1987) iscomputationally easy, and historically the most widelyused technique in determining the originating orbits ofmeteoroids. However, it does not allow forperturbations in orbit calculations such as third bodies(including the Moon) or Earth flattening effects. Anumerical approach is able to incorporate suchperturbations. With increasing computational power,such an approach is preferable.

A new numerical method (JS-19) is presented in thisstudy. To compare the results of this new orbitaldetermination technique to the typical analyticalmethod (C-87) and the numerical approach provided inthe Meteor Toolkit package (MT-15), the re-entryobservations of JAXA’s Hayabusa, with its knownheliocentric orbit as a “ground truth,” were invaluable.

As observations were made of both the spacecraft andthe capsule re-entry separately, these data provide twoexcellent test cases with which models could becompared to heliocentric telemetry. The spacecraft wasfirst observed at ~100 km altitude while the capsule wasnot observed until ~65 km altitude. The low observationaltitude of the capsule tests the capability of models toincorporate atmospheric influences. In both cases, JS-19determined the most similar orbit to JAXA’s recordedorbit than either C-87 or MT-15. This was especiallyevident in the second case due to the greateratmospheric influence that the capsule experiencedbefore initial sighting. Further investigation of theatmospheric influence shows the need for atmosphericconsideration in meteoroid orbit determination below~90 km altitude. This is therefore highly relevant formany meteorite dropping events which may not beinitially observed above this height by fireball networkstuned to brighter events. We also stressed that C-87alone does not account for atmospheric drag effects,requiring a pre-atmospheric initial velocity to bedetermined prior to its use. The calculation of thisinitial velocity by the majority of current fireballnetworks that use C-87 is unclear and may need to berevised.

We made a detailed assessment on the accuracy andprecision of orbital calculations. The numerical methodsare shown to produce more realistic precision anddeliver superior accuracy in estimating the Hayabusaspacecraft’s pre-Earth orbit from re-entry observationsthan the analytical method, verifying such claims ofprevious authors (Clark and Wiegert 2011; Jenniskenset al. 2011).

Fig. 8. The region of high eccentricity for the simulated data set of re-entry trajectories. (Color figure can be viewed at wileyonlinelibrary.com.)



The resulting orbital element precision is primarilydetermined by the size of the initial velocity magnitudeerror, as all other foreseeable uncertainties combinedcorrespond to orbital errors at least an order ofmagnitude smaller than the initial speed uncertainty, asdiscussed in the Precision section. While the precision ofthe orbit determination methods were comparable, JS-19 demonstrated greater accuracy due to its completedetailed representation of Earth’s gravity and itsinclusion of perturbations, as discussed in the Accuracysection.

By generating a great variety of simulated re-entrytrajectories, we were able to explore the effect ofdifferent perturbations by comparing orbits calculatedby both C-87 and JS-19. Simulated trajectories with lowentry velocities or which pass close to the Moon showthe most drastic orbital divergences. This demonstratesthe vital need for perturbation inclusion within the orbitdetermination method. The limitations of C-87 shouldbe considered and discussed if used for meteoroid orbitdetermination. Previously determined orbits, especiallythose in regions of significant orbital divergence (asdiscussed in the Hayabusa’s Orbit Determined from theCapsule’s Re-entry section) should be reanalyzed toavoid inaccurate orbital histories.

The Hayabusa case used in this work has provideda unique opportunity to compare orbit determinationtechniques. Although this case assesses only aheliocentric orbit, it must be noted that JS-19 cancompute an observed meteoroid’s orbit regardless ofwhether it originated around the Earth (geocentric),around the Sun (heliocentric), or from outside the solarsystem (hyperbolic), proving itself to be a more robustand real-world approach than its analytical counterpart.

Acknowledgments—We especially thank Dr. MakatoYoshikawa and the Hayabusa engineering team forproviding the telemetry data for the spacecraft that madethis study possible. We thank M. Gritsevich, J. Borovicka,and an anonymous reviewer for valuable feedback, whichhelped improve the quality of this manuscript. We alsothank the Australian Research Council for support undertheir Australian Discovery Project scheme. This work wasalso supported by an Australian Government ResearchTraining Program (RTP) Scholarship.

Editorial Handling—Dr. Josep M. Trigo-Rodr�ıguez

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APPENDIX: INITIAL CONDITIONS

The initial conditions for our comparative analysis are determined using the tabulated values given inBorovicka et al. (2011), but are collated in the table below. Note the quoted velocities are relative to the ground(ECEF frame).

Spacecraft Capsule

Epoch time 2010-06-13T13:51.56.6 2010-06-13T13:52.16.0Latitude (°) �29.0243 �29.6545Longitude (°) 131.1056 133.0768Height (km) 99.880 64.710Initial velocity (m s�1) 11,725.1 11,330.5Radiant azimuth (°) 290.5220 289.2733Radiant elevation (°) 10.0173 8.7955Mass (kg) 415 20Cross-sectional area (m2)a 2.15 0.126

Corresponding radius (m) 0.827 0.2Infinite velocity (m s�1)b 11,678.84 11,939.04a Using a drag coefficient of 2.b Using methods described in the appendix of Pecina and Ceplecha (1983).

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